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Operator algebraic approach to topological phases

Periodic Reporting for period 2 - OATP (Operator algebraic approach to topological phases)

Reporting period: 2017-09-01 to 2018-08-31

It was long thought that all phases of matter could be described by what is called Landau’s theory of symmetry breaking. With the discovery of topological phases in the early 1980’s, this belief turned out to be false. What distinguishes these new models is that fundamental physical characteristics are related to topological invariants of the system. Topology is the mathematical study of which shapes or geometric objects can be continuously deformed into each other, without “tearing” or breaking them. A topological invariant is then something that does not change under such a continuous deformation, such as the number of holes in a surface.

This robustness under deformations has interesting physical consequences, since it implies that corresponding physical properties are also robust. Hence these physical properties depend only qualitatively on the microscopic details of the system. Some of the most interesting applications of topological states of matter are in quantum computing. A quantum computer would provide huge benefits over ordinary “classical” computers in various computational tasks, such as factorisation of numbers into primes.

A major obstacle in building a quantum computer is that they typically are very sensitive to noise from the environment, which render their output useless. Although there are error correction protocols, it is tremendously difficult to make the error rate low enough. Therefore people have started looking for fault-tolerant quantum computers, which are inherently stable against noise from the environment. One idea is to use their topological properties for quantum computation. Since these topological properties are by their very nature stable against small perturbations of the system, this in principle can be used to provide fault tolerance.

Despite the many examples that have been found, there still is a lot that is unclear. Some properties are believed to be true based on physical grounds, but the underlying mathematical principles poorly understood. An example is the stability of such systems. Even though for specific systems there are good physics-based arguments on why they should be stable, a sound mathematical understanding is lacking. It is important to understand such fundamental properties better in a rigorous way, to be able to talk about many systems at the same time. In other words, to generalise these results to a wide class of systems.

In our project, we worked on the mathematical underpinnings of such systems. We focussed in particular on what are called quantum spin systems, which are defined on a lattice (or “grid”), with on each point on the grid a simple quantum spin. One can for example think of a solid, where atoms are localised in a regular manner. The interesting thing about the topologically ordered systems that we are interested in, is that they support what are called “anyons”. These are excitations of the system that behave like particles with very special properties (which again have a topological origin), and are used in proposals for fault-tolerant computing. In our project we showed that in certain models these anyons are indeed stable. That is, if we perturb the system a little bit, the structure of the anyons does not change.

Often it is difficult to extract all the properties of the anyons from a description of the system. Hence one is interested in finding invariants, which can be used to distinguish at least some models. In our work we found a new one, which can be defined using only a few natural assumptions.

The final objective of our project was to get a better physical understanding of the mathematics used to study anyons. It turns out that very abstract constructions there in fact have a very concrete interpretation in terms of quantum information. It can be shown that the so-called Jones index can be interpreted as the capacity of a wiretapping channel: the amount of information that can be send without an eavesdropper gaining any information. These results hint at powerful techniques in quantum information in infinite systems, which is important because we know that many physical systems are best described in this way.
The project has resulted in six publications, and an additional one nearing completion (see website for details). One of the publications is a book, providing graduate students and other researchers the necessary mathematical background of our work. It has already found use at various classes in different universities.

During the project we have investigated two main themes: stability of properties of topological phases (in particular anyons), and applications to quantum information theory. For the first topic, we managed to classify all ground states of abelian quantum double models, a particular model with topological order. Ground states are the lowest energy states and play a fundamental role in the theory. These ground states are related to the different types of anyons the theory has, and a good understanding of the ground states provide an essential first step in studying the stability of them. This is the main result of the project: a proof that the whole anyon structure of abelian quantum double models is stable: their properties do not change if we perturb the dynamics of the system (as long as the spectral gap does not close). We have also found a new invariant of certain topological phases. This invariant is related to the topological entanglement entropy in known cases, but does not suffer from some of its drawbacks.

The second topic is the application of the quantum dimension. We discovered that this quantity can be interpreted in terms of the amount of information one can hide from a malicious third party, using the anyons in a system. Also from a mathematical point of view this leads to very interesting structures, which open up new possibilities on the application of powerful advanced mathematical tools in quantum information theory.
Our work has provided a better understanding of the mathematics of topologically ordered systems. In particular the stability result is one the first of its kind. In addition, we have found a new invariant of a certain class of topological phases which can be defined rigorously for finite systems, making it amenable to numerical study. We have also gained a better understanding in the algebraic picture of systems with boundary. Clearly this is relevant in practice, since in an experiment all systems will be of finite size and hence have a boundary. This algebraic formulation makes it possible to start applying rigorous mathematical results on the stability of systems. It is clear that we are only at the beginning: our results could be seen as a proof of concept in simple examples, and have shown that to describe more realistic models, some of the assumptions have to be adapted, and hence has lead to many new questions.

Another exciting development is the use of subfactor theory in quantum information. Subfactor theory is well studied in mathematics, and has led to surprising connections between seemingly distinct topics in physics and mathematics. Our results provide one of the first examples of quantum information theory in systems with infinitely many degrees of freedom. Quantum information theory to infinite systems is important because we know that there are many systems in nature that are best described as infinite systems. We expect this to become an important topic in the near future.